Conditional probabilities for Euro area sovereign default risk

Transcription

1 Conditional probabilities for Euro area sovereign default risk Andre Lucas, Bernd Schwaab, Xin Zhang Rare events conference, San Francisco, Sep webpage: Disclaimer: Not necessarily the views of ECB or ESCB.

2 Contributions We propose a novel modeling framework to infer conditional and joint probabilities for sovereign default risk from observed CDS. Novel framework? Based on a dynamic GH skewed t multivariate density/copula with time-varying volatility and correlations. Multivariate model is suffi ciently flexible to be calibrated daily to credit market expectations. Not an "offi cial opinion". Analysis is based on Euro area CDS data from 2008M1 to 2011M6. Event study: SMP/EFSF announcement & initial impact on risk.

3 Literature 1. Sovereign credit risk: e.g. Pan and Singleton (2008), Longstaff, Pan, Pedersen, and Singleton (2011), Ang and Longstaff (2011). 2. Contagion, see e.g. Forbes and Rigobon (2002), Caporin, Pelizzon, Ravazzolo, Rigobon (2012). 3. Observation-driven time-varying parameter models, see Creal, Koopman, and Lucas (2011, 2012), Zhang, Creal, Koopman, Lucas (2011), Creal, Schwaab, Koopman, Lucas (2011), Harvey (2012). 4. Non-Gaussian dependence/copula/credit modeling, see e.g. Demarta and McNeil (2005), Patton and Oh (2011).

4 Empirical questions (Q1) Financial stability information: Based on credit market expectations, what is... Pr(two or more credit events in Euro area)? Pr(i j)-pr(i), for any i,j? Spillovers, e.g. Pr(PT GR) - Pr(PT not GR)? Corr t (i,j) at time t? (Q2) Model risk: For answering (a), how important are parametric assumptions? Normal vs Student-t vs GH skewed-t. (Q3) Event study: did the May 09, 2010 Euro area rescue package change risk dependence? How?

5 The GHST copula framework Sovereign defaults iff benefits (v it ) exceed a cost (c it ), where v it = (ς t µ ς ) L it γ + ς t L it ɛ t, i = 1,..., n, ɛ t N(0, I n ) is a vector of risk factors, L it contains risk factor loadings, γ R n determines skewness, ς t IG is an additional scalar risk factor for, say, interconnectedness. A default occurs with probability p it, where p it = Pr[v it > c it ] = 1 F i (c it ) c it = F 1 i (1 p it ), where F i is the CDF of v it. Focus on conditional probability Pr[v it > c it v jt > c jt ], i = j.

6 Data: skewed, fat tailed, tv vol s and correlation 2500 AT BE DE ES FR GR 1500 IE IT NL PT 500 Austria Greece squared differences, AT squared differences, GR Average of rolling window correlations

7 The GH skewed-t multivariate distribution y t = µ + L t e t, t = 1,..., T, e t GHST, E[e t e t] = I n, p(y t ; ) = Γ ( υ 2 υ 2 υ υ+n 21 2 Kυ+n 2 ) n 1 π 2 Σ t 2 ( d(yt ) (γ γ)) e γ L 1 t (y t µ t ) (d(y t ) (γ γ)) υ+n 4 d(y t ) υ+n 2, where d(y t ) = υ + (y t µ t ) Σ t 1 (y t µ t ), µ t = υ/(υ 2) L t γ, Σ t = L t L t is scale matrix If γ = 0, then GH skewed-t simplifies to Student s t density. If in addition υ 1 0, then multivariate Gaussian density. Σ t (f t ) = L t (f t ) L t (f t ) is driven by 1st and 2nd derivative of the pdf.

8 The model with time varying parameters Assume that Σ t = D t R t D t = L t (f t )L t (f t ) and that f t+1 = ω+ p 1 i=0 A i s t i + q 1 j=0 B jf t j, where s t = S t t is the scaled score t = ln p(y t ; Σ(f t ), γ, υ)/ f t S t = E t 1 [ t t y t 1, y t 2,...] 1, Scaling matrix S t is inverse conditional Fisher information matrix.

9 Time varying parameters: score Important: first two derivatives are available in closed form. t = ln p(y t ; Σ(f t ), γ, υ)/ f t = vech(σ t) vech(l t ) f t vech(σ t ) =... = Ψ th t vec {w t y t y t Σ t where Ψ t = vech(σ t )/ f t H t = messy w t = υ + n k 2 d(y t ) v +n 2 vec( L t ) ln p GH (y t f t ) vech(l t ) vec( L t ) ( d(yt ) (γ γ)) d(yt )/γ γ ( 1 υ ) } υ 2 w t L t γy t ; k a(b) = ln K a(b). b

10 Extracting marginal pd s from CDS We equate the premium and default leg of CDS given a default intensity. where p it s it(1 + r t ) 1 rec i, ( ) s it = CDS annual fee, country i, time t r t = LIBOR 1 year rate, flat rec i = 25% expected recovery, stressed. Eqn ( ) is exact if the term structures for pd s and interest rates are flat, s it is paid to the seller continuously, and there is no counterparty credit risk, see Brigo and Mercurio (2007).

11 Marginal pd s from CDS Austria Germany France Ireland Netherlands Belgium Spain Greece Italy Portugal

12 Volatility estimates 0.15 Austria CDS changes squared Austria t Est. vol Germany CDS changes squared Germany t Est. vol France CDS changes squared 0.02 France t Est. vol Austria Gaussian Est. vol Austria GHST Est. vol Belgium CDS changes squared Belgium t Est. vol Germany Gaussian Est. vol Spain CDS changes squared 2 Germany GHST Est. vol Spain t Est. vol 1 15 France Gaussian Est. vol France GHST Est. vol 5 Greece CDS changes squared Greece t Est. vol Belgium Gaussian Est. vol Belgium GHST Est. vol Spain Gaussian Est. vol Spain GHST Est. vol Greece Gaussian Est. vol Greece GHST Est. vol Ireland CDS changes squared 0.50 Ireland t Est. vol Ireland Gaussian Est. vol Italy CDS changes squared 0.50 Ireland GHST Est. vol Italy t Est. vol Italy Gaussian Est. vol Italy GHST Est. vol Netherlands CDS changes squared Netherlands t Est. vol Netherlands Gaussian 3 Est. volportugal CDS changes squared Netherlands GHST Est. vol Portugal t Est. vol 1 Portugal Gaussian Est. vol Portugal GHST Est. vol

13 Dynamic correlations Gaussian correlation Rolling window correlation t correlation Rolling window correlation GHST correlation Rolling window correlation 0.50 Gaussian correlation t correlation GHST correlation rolling window correlation

14 The probability of two or more failures Prob of 2 or more defaults, Gaussian Symmetric t GHST

15 The probability of k=0,1,2,... failures default 1 default 2 defaults 3 defaults 4 defaults 5 defaults 6 defaults 7 defaults 8 defaults 9 defaults 10 defaults

16 Conditional pds: Pr(all i all j) Austria Belgium Germany Spain France Greece Ireland Italy Netherlands Portugal

17 Conditional pds: Pr(i GR) 0.6 Gaussian 0.6 Symmetric t GH Skewed t Austria GH Skewed t Belgium Germany with zero correlation Spain France Ireland Italy Netherlands Portugal

18 GHST spillovers: Pr(i GR) - Pr(i not GR) 0.75 Pr( i GR) Pr( i not GR) Austria Belgium Germany Spain 0.03 France Ireland Italy Netherlands 0.02 Portugal Pr( i GR) Pr( i not GR) 0.50

19 The May 09, 2010 package Joint risk, Pr(i j) Thu 06 May 2010 Tue 11 May 2010 PT GR DE PT GR DE AT 1.1% 1.1% 0.6% 0.6% 0.7% 0.4% BE 1.2% 1.4% 0.7% 0.9% 1.0% 0.6% DE 1.0% 1.1% 0.8% 0.8% ES 3.0% 3.3% 0.9% 1.5% 1.6% 0.6% FR 1.0% 1.0% 0.6% 0.8% 0.9% 0.6% GR 4.8% 1.1% 2.3% 0.8% IR 2.6% 3.1% 0.8% 1.4% 1.8% 0.6% IT 2.8% 2.9% 0.9% 1.4% 1.5% 0.6% NL 0.9% 0.9% 0.5% 0.6% 0.7% 0.5% PT 4.8% 1.0% 2.3% 0.8% Avg 2.0% 2.2% 0.8% 1.1% 1.2% 0.6%

20 The May 09, 2010 package Conditional risk, Pr(i j) Thu 06 May 2010 Tue 11 May 2010 PT GR DE PT GR DE AT 17% 8% 53% 22% 10% 46% BE 20% 10% 60% 32% 15% 61% DE 16% 8% 26% 12% ES 49% 25% 78% 50% 23% 63% FR 16% 8% 58% 28% 12% 62% GR 78% 99% 80% 86% IR 43% 23% 75% 49% 26% 68% IT 45% 22% 77% 49% 21% 64% NL 14% 7% 49% 21% 10% 50% PT 36% 91% 33% 81% Avg 33% 16% 71% 40% 18% 64% Bottom line: joint risks, but dependence. "Firewall"-analogy?

21 Conclusion We propose a novel modeling framework to infer conditional and joint probabilities for sovereign default risk from observed CDS. Novel framework? Based on a dynamic GH skewed t multivariate density/copula with time-varying volatility and correlations. Multivariate model is suffi ciently flexible to be calibrated daily to credit market expectations. Analysis is based on Euro area CDS data from 2008M1 to 2011M6. Event study: SMP/EFSF announcement & initial impact on risk.

22 Thank you